Class 3c: Review of concepts in Probability and Statistics

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# Class 3c: Review of concepts in Probability and Statistics
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.author[
### Business Forecasting
]

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# Hypothesis Testing

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### Hypothesis Testing

Let's go back to our question. Can we get a higher price if cleanliness level is above 4.5?

This type of questions can be answered with hypothesis testing where we try to see which one of competing hypothesis is more likely to be true.

- .blue[Hypothesis] is a claim about a parameter or a distribution.

--
1. `$H_0$` **Null Hypothesis**: 
  - Claim to be tested, skeptical perspective

--
      - Prices of clean and dirty apartments are equal 
      - `$H_0: \mu_c=\mu_d$`

--
2. `$H_A$` **Alternative Hypothesis**:
  - An alternative idea under consideration

--
      - Prices of clean apartments are higher than prices of dirty apartments
       - `$H_A: \mu_c>\mu_d$`

- Assume that the null is true and see if there is enough of data to reject the null in favor of the alternative.
- You either reject the null, or fail to reject it (but not accept it)
---
### Hypothesis Testing

Some examples of hypotheses testing:

- You released a new promotion to some customers (10% off) and you want to test if people who got the promotion spend different amount of money than those who didn't

--
  - `$H_0$`: People who got the promotion spend the same as people who didn't get the promotion
      - `$H_0: \mu_p =  \mu_n$`
  - `$H_A$`: People who got the promotion spend different amount than people who didn't get the promotion
      - `$H_A: \mu_p \neq \mu_n$`

- You designed a new healthy snack and want to test whether it needs "excessive sugar" sticker. It gets the sticker if it exceeds 100g of sugar. You take a sample and test.

--
  - `$H_0$`: The average sugar content is 100 or less
      - `$H_0: \mu_s =  100$`
  - `$H_A$`: The average sugar content is more than 100
      - `$H_A: \mu_s >  100$`

---
### Hypothesis Testing

- Suppose we calculated the sample mean for sugar content. It's either below or above 100g

- Why would we need a fancy test?

- Because we only have a sample, but we want to learn about the population parameter!

- Maybe in our sample it's below 100g, but in the whole population it's above

- Hypothesis are always about parameters! Never about sample sample statistics

---
### Testing procedure

**Test procedure** is a rule based on a sample data whether to reject the null or not.
 
It usually relies on two elements:
 - .blue[Test statistics] 
  - Function of the sample data summarizing evidence for or against a hypothesis 
  - Usually we know how it should be distributed under the null. Ex: standardized mean
 - .blue[Rejection region] 
  - Values of the test statistics such that we would reject the null if we observe them
    - Basically, values which are very unlikely under the null hypothesis
  - Typically threshold based:
  - Such as: reject `$H_0$` if the test statistic > k, where k is some threshold

---
### Testing procedure: Example

Let's go back to the sugar example. Suppose we have a sample `$n=36$`.

- .blue[Test statistics] 
  - Our test statistic could be:
  `$$z=\frac{\bar{x}-100}{\underbrace{s_x/\sqrt 36}_{SE}}$$`

- .blue[Rejection region] 
 - For a 5% significance level, our rejection region would be: 
 - `$\{1.645, \infty \}$`
 
.center[
<img src="data:image/png;base64,#C_3_slides_c_files/figure-html/unnamed-chunk-1-1.png" width="284" />
]

---
### Testing procedure: Example

- Suppose `$\sigma_x=20$`. When would we reject?

Reject if: 
- `$\frac{\bar{x}-100}{20/\sqrt 36}>1.645$`
- Or equivalently if `$\bar{x}>105.483$`
- Those two formulations are fine 
--

- Why don't we reject even if `$\bar{x}=103$`?

---
### Testing procedure: Example
- Because there could be randomness in the sample:

- So even if the `$\mu_s=100$` we could get a sample with `$\bar{x}>100$`

.center[
<img src="Plot_sampl_dist.png" style="width:70%">
]

- Why `$\bar{x}>105.483$` though? 
- Formally about it later, but it's very unlikely to get a sample mean that large if `$\mu_s=100$`

---

### Decision errors

- Suppose that the null is true and `$\mu_s=100$`

--
- You take a sample of 36.

--
- But you were very unlucky in your sampling, so by chance you actually got a sample with those rare packages with  a lot of sugar and your `$\bar{x}=106$`

--
- According to the test procedure, you reject the null

--
- You make an error

--
- What's the probability of making such error?

`$$\alpha=P(\text{Type 1 error})=\underbrace{P_{H_0}(\bar{X}>105.483)}_{\text{Pr. if null is true}}=P(Z>1.645)=1-\underbrace{\Phi(1.645)}_{\text{Normal CDF}}$$`

---

### Decision errors

|             | Do not reject `$H_0$` | Reject `$H_0$` in favor of `$H_A$` |
| ----------- | ------------------- | ------------------------------- |
| `$H_0$` true  | Okay                | Type 1 Error                    |
| `$H_A$` true  | Type 2 Error        | Okay                            |

- `$H_0$`: You are not pregnant
- `$H_1$`: You are pregnant

--
.center[
<img src="pregnancy.jpg" style="width:50%">
]
---
### Decision errors

Do prople spend more if they get free shipping? ($H_0$ is that they spend the same)

- .blue[Type 1 error]: reject null when `$H_0$` is true
  - Conclude that people spend more when they got free shipping, while in reality they spend the same as those who didn't

--
- .blue[Type 2 error]: don't reject null when `$H_A$` is true
  - Conclude that people who got free shipping spend the same as those without free shipping, while those with free shipping spend more

--
- Imagine a trial of a murderer and suppose `$H_0$` is innocent, and `$H_1$` is guilty. 
- Describe error type 1 and 2 in these scenarios

--
  - Type 1: Convict an innocent
  - Type 2: Do not convict a murderer

--
- Probability of making type 1 error is `$\alpha$`
  - It's also called .blue[size], or .blue[significance level] of the test
  - We don't want to incorrectly reject null more than `$\alpha$`*100 percent of times.

--
- Probability of making type 2 error is `$\beta$`
  - To calculate `$\beta$`, you need to know the true `$\mu$`

---
### Probability of type 2 error
- In our example (and valid only for this example): `$\mu_s$` is true sugar content
- We reject if `$\scriptsize z=\frac{\bar{x}-100}{\sigma_x/ \sqrt 36}>1.645$`
- Or equivalently if `$\scriptsize \bar{X}>105.483=\mu_0+\underbrace{z_{\alpha}}_{Crit.Val:1.645}*\frac{\sigma_x}{\sqrt n}$`

- Type 2 error is if we don't reject when alt. is true
`$$\scriptsize \beta=P(\text{Type 2 error})=\underbrace{P_{H_A}(\bar{X}<105.483)}_{\text{Pr. if alt is true}}=\underbrace{P_{H_A}(\bar{X}<\mu_0+z_{\alpha}\frac{\sigma_x}{\sqrt n})}_{\text{Pr. if alt is true}}=P(\frac{\bar{X}-\mu_s}{\sigma_x/ \sqrt n}<\frac{\mu_0+z_{\alpha}\frac{\sigma_x}{\sqrt n}-\mu_s}{\sigma_x/ \sqrt n})$$`
Rearranging and applying to our example
`$$\scriptsize \beta=P(Z<\frac{\mu_0-\mu_s}{\sigma_x/ \sqrt n}+z_{\alpha})=\underbrace{\Phi(\frac{\mu_0-\mu_s}{\sigma_x/ \sqrt n}+z_{\alpha})}_{\text{Normal CDF}}==\underbrace{\Phi(\frac{100-\mu_s}{20/ \sqrt{36}}+1.645\frac{20}{\sqrt 36})}_{\text{Normal CDF}}=\underbrace{\Phi(\frac{105.483-\mu_s}{20/ \sqrt{36}})}_{\text{Normal CDF}}$$`

- depends on what true `$\mu$` is and where our rejection region is!
- When you decrease probability of error type 1, you increase the probability of error type 2

---
### Trade-off between type 1 and 2 errors

`$$\beta=P(\text{Type 2 error}) \beta=P(Z<\frac{\mu_0-\mu_s}{\sigma_x/ \sqrt n}+z_{\alpha})=\underbrace{\Phi(\frac{\mu_0-\mu_s}{\sigma_x/ \sqrt n}+z_{\alpha})}_{\text{Normal CDF}}$$`

`$$\alpha=P(\text{Type 1 error})=P(Z>z_{\alpha})=1-\underbrace{\Phi(z_{\alpha})}_{\text{Normal CDF}}$$`
If I change critical value `$z_{\alpha}$`, it will have the opposite effect on them.

My app or:
Source: https://shiny.rit.albany.edu/stat/betaprob/

---
### Power of a test

- Power of a test is the probability of rejecting if alternative is true
- Power will be different for any possible value of the alternative
- Notice that

`$$Power=P_{H_A}(Reject)=1-\underbrace{P_{H_A}(\text{Not Reject})}_{\text{type 2 error}}=1-\beta$$`

Source: https://shiny.rit.albany.edu/stat/betaprob/

---
### Power of a test 
- Let's go back to sugar example. Our rule was reject `$H_0$` if `$z>1.645$`. 
- Suppose `$\mu_s=110$` and `$\sigma=20$`. Let's calculate power of the test:

`$$\scriptsize P_{H_A}(Z_{test}>1.645)=P_{H_A}(\frac{\bar{X}-100}{\underbrace{ \sigma_x/ \sqrt 36}_{SE}}>1.645)=P_{H_A}(\frac{\bar{X}-\mu_0}{\underbrace{ \sigma_x/ \sqrt n}_{SE}}>z_{\alpha})=P_{H_A}(\bar{X}>\mu_0+z_{\alpha}\frac{\sigma_x}{\sqrt n})=P_{H_A}(\bar{X}>105.483)$$`
--

- To calculate this probability, let's standarize `$\bar{X}$`:
  - We subtract the true mean of the distribution and the standard deviation

`$$\scriptsize P_{H_A}(\frac{\bar{X}-\mu_s}{ \sigma_x/ \sqrt n}>\frac{\mu_0-\mu_s}{\sigma_x/ \sqrt n}+z_{\alpha})=P_{H_A}(\frac{\bar{X}-\mu_s}{ \sigma_x/ \sqrt n}>\frac{-10}{20/ \sqrt 36}+1.645)=P_{H_A}(\underbrace{\frac{\bar{X}-\mu_s}{ \sigma_x/ \sqrt n}}_{st.normal}>-1.355)=0.9115$$`
Then: `$\scriptsize \beta=1-0.9115=0.0885$`
 
--

Power increases if:
- n increases
- Alternative is further away from the null
- `$\sigma$` decreases
- `$\alpha$` increases

---

### General procedure:

1. Determine the the null and the alternative hypothesis

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2. Collect your sample of independent observations

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3. Choose the appropriate test

--
4. Calculate test statistic

--
5. Compare it to the rejection region

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5. Reject or not your null hypothesis

--
Let's talk about **appropriate tests**!
---

### Appropriate tests

Choice of the test statistic and rejection region will depend on:

- **Type of data we are testing?**
  - .blue[Single sample]
      - Example of hypothesis: `$H_0: \mu=\mu_0$` vs `$H_A: \mu \neq \mu_0$` 
  - .blue[Two independent samples]
      - Example of hypothesis: `$H_0: \mu_{s1}=\mu_{s2}$` vs `$H_A: \mu_{s1} \neq \mu_{s2}$`
  - .blue[Paired data]
      - Example of hypothesis: `$H_0: \mu_{d1}=\mu_{d2}$` vs `$H_A: \mu_{d1} \neq \mu_{d2}$`

- **Can we estimate standard deviation well?**
  - .blue[Yes]
      - Large sample
      - Known standard deviation
  - .blue[No]
      - Small sample but normal distribution

---

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# Hypothesis Testing: Single Sample

---

### Single Sample Test for Mean

**General idea:** 
- We test if the parameter is equal/larger/smaller than some concrete value
  - Ex: `$H_0: \mu=3$` vs `$H_A: \mu \neq 3$`

--
- .blue[Test statistic]: normalized sample mean

`$$\text{test statistic}=\frac{\bar{X}-\mu_0}{SE}$$`
 
  - Where SE is standard error and depends on estimate of standard deviation
      - If standard deviation known: `$\small SE=\frac{\sigma}{\sqrt n}$`
      - If standard deviation not known: `$\small SE=\frac{s}{\sqrt n}$`

--
- .blue[Rejection region]: depends on the distribution 
  - If large sample or known variance `$\small \text{test statistic}=Z_{test} \sim N(0,1)$`
      - Critical values come from standard normal distribution
  - If small sample with normal distribution `$\small \text{test statistic}=T_{test} \sim t(n-1)$`
      - Critical values come from student t with n-1 degrees of freedom

-
---
### Single Sample - Mean - Two Sided Test

**Hypothesis:**
`$H_0: \mu=\mu_0$`  and 
`$H_A: \mu \neq \mu_0$`

**Rejection Region:**
For a test of significance level `$\alpha$`, reject

- If large sample or known variance from normal
`$$\small \frac{\bar{X}-\mu_0}{SE}<-z_\frac{\alpha}{2} \qquad or \qquad \frac{\bar{X}-\mu_0}{SE}>z_\frac{\alpha}{2}$$`
--

- If small sample from normal
`$$\small \frac{\bar{X}-\mu_0}{SE}<-t_{(n-1),\frac{\alpha}{2}} \qquad or \qquad \frac{\bar{X}-\mu_0}{SE}>t_{(n-1),\frac{\alpha}{2}}$$`
--
Where 
- `$z_\frac{\alpha}{2}$` and `$t_{(n-1),\frac{\alpha}{2}}$` are `$\small (1-\frac{\alpha}{2})$` quantiles of standard normal and t-students with n-1 degrees of freedom

---
### Single Sample - Mean - Two Sided Test

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---
### Link between Tests and Confidence Intervals

Imagine a test for a parameter `$\mu$` with hypothesis `$H_0=4$` and `$H_A \neq 4$`.

- Suppose you draw a sample of 49 units with a mean `$\bar{x}=3$` and standard deviation is `$s=3$`
- Our test statistic is `$z_{test}=\frac{3-4}{3/7}=-2.34$`
- Our critical values at `$\alpha=0.05$` are -1.96 and 1.96, hence we reject

--
- We can also calculate the 95% confidence interval for our sample mean:

`$$\small \{3-1.96\frac{3}{\sqrt 49}, 3+1.96\frac{3}{\sqrt 49}\}=\{2.16, 3.84\}$$`
- It does not contain the null hypothesis

---
### Link between two sided Tests and Confidence Intervals

- More generally: the `$1-\alpha$` confidence interval doesn't contain null = null would be rejected with a test of significance `$\alpha$`

https://kzaremba.shinyapps.io/Hypothesis_Confidence/

--
- Mathematically:

- Let `$\mu_0$` be the null hypothesis. We reject (in two sided test) the null at `$\alpha$` if 
  `$$\scriptsize \left| \small \frac{\bar{X}-\mu_0}{ s/ \sqrt n} \right|>z_{\frac{\alpha}{2}}$$`
  
  - Which can be rewritten as:

`$$\scriptsize \mu_0<\bar{X}-z_{\frac{\alpha}{2}}\frac{s}{\sqrt n} \qquad or \qquad \mu_0>\bar{X}+z_{\frac{\alpha}{2}}\frac{s}{\sqrt n}$$`

---

### Single Sample - Mean - One sided - Case 1

**Hypothesis:**
`$H_0: \mu=\mu_0$` and 
`$H_A: \mu < \mu_0$`

- This includes any null hypothesis like `$\mu \geq \mu_0$`. Why?

--
- If you rejected `$\mu_0$` at `$\alpha$`, you would reject for sure anything larger than `$\mu_0$`

--
**Rejection Region:**
For a test of significance level `$\alpha$`, reject

- If large sample or known variance from normal
`$$\small \frac{\bar{X}-\mu_0}{SE}<-z_\alpha$$`
--

- If small sample from normal
`$$\small \frac{\bar{X}-\mu_0}{SE}<-t_{(n-1),\alpha}$$`
--
Where 
- `$z_\alpha$` and `$t_{(n-1),\alpha}$` are `$\small (1-\alpha)$` quantiles of standard normal and t-students with n-1 degrees of freedom

---
### Single Sample - Mean - One sided - Case 1

.center[
<img src="data:image/png;base64,#C_3_slides_c_files/figure-html/unnamed-chunk-4-1.png" width="100%" />
]

- .blue[Intuition]: 
 - We never reject `$H_0$` if `$\bar{X} \geq \mu_0$`.
 - We reject `$H_0$` if `$\bar{X}$` is sufficiently smaller than `$\mu_0$`
 - If null is true ( `$\mu=\mu_0$` ) then we reject with probability
`$$\small P(\frac{\bar{X}-\mu_0}{s/\sqrt n}<-z_\alpha)=P(Z<-z_\alpha)=\alpha$$`

---

### Single Sample - Mean - One sided - Case 2

**Hypothesis:**
`$H_0: \mu=\mu_0$` and 
`$H_A: \mu > \mu_0$`

- This includes any null hypothesis like `$\mu \leq \mu_0$`. Why?

--
- If you rejected `$\mu_0$` at `$\alpha$`, you would reject for sure anything smaller than `$\mu_0$`

--
**Rejection Region:**
For a test of significance level `$\alpha$`, reject

- If large sample or known variance from normal
`$$\small \frac{\bar{X}-\mu_0}{SE}>z_\alpha$$`
--

- If small sample from normal
`$$\small \frac{\bar{X}-\mu_0}{SE}>t_{(n-1),\alpha}$$`
--
Where 
- `$z_\alpha$` and `$t_{(n-1),\alpha}$` are  `$\small (1-\alpha)$` quantiles of standard normal and t-students with n-1 degrees of freedom

---
### Single Sample - Mean - One sided - Case 2

.center[
<img src="data:image/png;base64,#C_3_slides_c_files/figure-html/unnamed-chunk-5-1.png" width="100%" />
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- .blue[Intuition]: 
  -  We never reject `$H_0$` if `$\bar{X} \leq \mu_0$`.
  -  We reject `$H_0$` if `$\bar{X}$` is sufficiently larger than `$\mu_0$`
  -  If null is true ( `$\mu=\mu_0$` ) then we reject with probability
`$$\small P(\frac{\bar{X}-\mu_0}{s/\sqrt n}>z_\alpha)=P(Z>z_\alpha)=\alpha$$`

---
### Tests for Variance
Test for variance has the same idea, but different distribution of the test statistic.
Suppose you have an IID sample from .blue[normal distribution]

**Test statistic** for `$H_0: \sigma=\sigma_0$`  and its distribution if null is true is:
`$$\small \chi_{test}=  \frac{(n-1)S^2}{\sigma _{0}^{2}}\sim \chi_{n-1}$$`
It is distributed as a Chi-square with n-1 degrees of freedom if null is true:

--
**Rejection regions:**
1. `$\small H_A: \sigma \neq\sigma_0$`
 - Reject null if `$\small \chi_{test}<\chi_{\frac{\alpha}{2},n-1}$` or `$\small \chi_{test}>\chi_{1-\frac{\alpha}{2},n-1}$`

--
2. `$\small H_A: \sigma \leq \sigma_0$`
 - Reject null if `$\small \chi_{test}<\chi_{\alpha,n-1}$`

--
3. `$\small H_A: \sigma \geq \sigma_0$`
  - Reject null if `$\small \chi_{test}>\chi_{1-\alpha,n-1}$`

--
Where:
- `$\small \chi_{\alpha,n-1}$` is `$\small \alpha$` quantile of `$\chi_{n-1}$`
- `$\small \chi_{1-\alpha,n-1}$` is `$\small 1-\alpha$` quantile of `$\chi_{n-1}$`
- `$\small \chi_{df}$` distribution is not symmetric around 0!

---

**Exercise**

Suppose you are a ham producer. As you need to stick to the nutritional guidelines, the standard deviation of the fat content in your product needs cannot be larger than 0.1. You take a sample of 16 hams and you want to test this. Suppose `$s=0.2$` and `$\alpha=0.05$`

--
1. Write down the hypotheses of the test

--
- Remember test statistic is about variance

--
2. What is type 1 and type 2 error in this context

--
3. Test the claim at 5% significance level. State assumptions you need to implement it

--
4. What is 95% confidence interval for variance
---

### P-values

**Technically:**
The P-value is the probability (so between 0 and 1), under the null hypothesis, of obtaining a value of the test statistic at least as contradictory to H0 as the value calculated from the available sample.

Depends on:
- Your hypothesis
- Calculated test statistic

--
- Suppose the test statistic under `$H_0$` is `$Z_{test} \sim N(0,1)$`
- One sided tests:
 - For `$H_A: \mu>\mu_0$` `$p-value=P(Z_{Test} \geq (computed-test-statistic))$`
 - For `$H_A: \mu<\mu_0$` `$p-value=P(Z_{Test} \leq (computed-test-statistic))$`
- Two sided test:
 - For `$H_A: \mu \neq\mu_0$` `$p-value=2P(Z_{Test} \geq |(computed-test-statistic)|)$`
---
**Example**: 
- We test for `$H_0: \mu=5$` and `$H_A: \mu>5$` 
- Suppose in sample a of 36 we have `$\bar{x}=5.5$` and `$s=2$`.

--
- Test statistic in this case is distributed normally
`$$p-value=P(Z>\frac{5.5-5}{2/6})=P(Z>1.5)=0.066$$`

**Intuitively:**
The probability that a statistic we calculated (or more extreme value) could arise just by chance if null is true

---
### P-value in two-sided test

---
### P-Values and test significance

- In our example we calculated the p-value of 0.066
- Would the test at `$\alpha=0.05$` reject the null?

--
- Any test with the significance level `$\alpha>p-value$` would reject the null

**Another way to think about p-values**
- The P-value is the smallest significance level `$\alpha$` at which the `$H_0$`
can be rejected

---
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# Hypothesis Testing: Two samples

---

### Two Samples and the Difference in Means
We have 2 iid samples .red[independent] of each other from two populations `$X_1,X_2,..,X_n$` and `$Y_1,Y_2,..,Y_m$`.

**General idea:** 
- We test the difference between .blue[population means] in two populations
  - Ex: `$H_0: \mu_X-\mu_Y=100$` vs `$H_A: \mu_X-\mu_Y \neq 100$`
  - Ex: `$H_0: \mu_X-\mu_Y=0$` vs `$H_A: \mu_X-\mu_Y>0$`
  
--
  - Test for no difference:
      - Ex: `$H_0: \mu_X-\mu_Y=0$` vs `$H_0: \mu_X-\mu_Y \neq 0$`

--
- .blue[Test statistic]: normalized difference in sample means
`$$\text{test statistic}=\frac{\bar{X}-\bar{Y}-\overbrace{(\mu_{X,0}-\mu_{Y,0})}^{\Delta_0}}{SE}=\frac{\bar{X}-\bar{Y}-\Delta_0}{SE}$$`
 
---
### Variance in two samples

- Where SE is standard error and depends on the estimate of standard deviation
      - If standard deviation known: `$\small SE=\sqrt{\frac{\sigma_X^2}{n}+\frac{\sigma_Y^2}{m}}$`

- Why? Because `$\small Var(\bar{X}-\bar{Y})=Var(\bar{X})+Var(\bar{Y})=\frac{\sigma_X}{n}+\frac{\sigma_Y}{m}$`
  - If standard deviation not known: `$\small SE=\sqrt{\frac{s_X^2}{n}+\frac{s_Y^2}{m}}$`

- .blue[Rejection region]: depends on the distribution 
  - If large sample or known variance from normal `$\small \text{test statistic}=Z_{test} \sim N(0,1)$`
      - Critical values come from standard normal distribution
  - If small sample with normal distribution `$\small \text{test statistic}=T_{test} \sim t(v)$`
      - Critical values come from student t with v degrees of freedom

---

**Side note on the number of degrees of freedom for difference in means**

Two ways to do it:

--
1. Proper, complex (and annoying):

`$$\frac{\left(\frac{s_X^2}{n}+\frac{s_Y^2}{m}\right)^2}{ \left(\frac{s_X^2}{n} \right)^2/(n-1)+ \left(\frac{s_Y^2}{m} \right)^2/(m-1)}$$`
Round down to nearest integer

--
2. Simple, approximate shortcut:

$$v=min(n-1,m-1) $$

---
### Difference in means - Two Sided Test

**Hypothesis:**
`$H_0: \mu_x-\mu_Y=\Delta_0$`  and 
`$H_A: \mu_x-\mu_Y \neq \Delta_0$`

**Rejection Region:**
For a test of significance level `$\alpha$`, reject

- If large sample or known variance from normal
`$$\small \frac{\bar{X}-\bar{Y}-\Delta_0}{SE}<-z_\frac{\alpha}{2} \qquad or \qquad \frac{\bar{X}-\bar{Y}-\Delta_0}{SE}>z_\frac{\alpha}{2}$$`
--

- If small sample from normal
`$$\small \frac{\bar{X}-\bar{Y}-\Delta_0}{SE}<-t_{v,\frac{\alpha}{2}} \qquad or \qquad \frac{\bar{X}-\bar{Y}-\Delta_0}{SE}>t_{v,\frac{\alpha}{2}}$$`
--
Where 
- `$z_\frac{\alpha}{2}$` and `$t_{v,\frac{\alpha}{2}}$` are `$\small (1-\frac{\alpha}{2})$` quantiles of standard normal and t-students with v degrees of freedom

---
### Difference in means - Two Sided Test

If CLT kicks in, the distributions are:

`$$\bar{X} \sim N(\mu_X, \sigma_X/\sqrt n) \qquad and \qquad \bar{Y} \sim N(\mu_Y, \sigma_Y/\sqrt m)$$`
The difference of two normal (independent) distributions under:

`$$\bar{X}-\bar{Y} \sim N(\mu_X-\mu_Y, \sqrt{\sigma_X^2/n+\sigma_Y^2/m})$$` 
So if null is true, then standardized statistic:

`$$\frac{\bar{X}-\bar{Y}-(\mu_{X,0}-\mu_{Y,0})}{\sqrt{s_X^2/n+s_Y^2/m}} \sim N(0,1)$$`
---

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### Confidence interval for a difference

Following this idea, we can construct a confidence interval for the difference:

`$$\small P(-z_{\alpha/2}<\frac{\bar{X}-\bar{Y}-(\mu_{X,0}-\mu_{Y,0})}{\sqrt{s_X^2/n+s_Y^2/m}}<z_{\alpha/2})=1-\frac{\alpha}{2}$$`
`$$\small P(\bar{X}-\bar{Y}-z_{\alpha/2}\sqrt{s_X^2/n+s_Y^2/m}<\mu_{X,0}-\mu_{Y,0}<\bar{X}-\bar{Y}+z_{\alpha/2}\sqrt{s_X^2/n+s_Y^2/m})=1-\frac{\alpha}{2}$$`

`$$\small CI_{\alpha}=(\bar{X}-\bar{Y} \pm z_{\alpha/2}\sqrt{s_X^2/n+s_Y^2/m})$$`

---

### Two Samples - Mean - One sided - Case 1

**Hypothesis:**
`$H_0: \mu_X-\mu_Y=\Delta_0$` and 
`$H_A: \mu_X-\mu_Y<\Delta_0$`

- This includes any null hypothesis like `$\mu_X-\mu_Y \geq \Delta_0$`. Why?

--
- If you rejected `$\mu_X-\mu_Y=\Delta_0$` at `$\alpha$`, you would reject for sure anything larger than `$\Delta_0$`

--
**Rejection Region:**
For a test of significance level `$\alpha$`, reject

- If large sample or known variance from normal
`$$\small \frac{\bar{X}-\bar{Y}-(\Delta_0)}{SE}<-z_\alpha$$`
--

- If small sample from normal
`$$\small \frac{\bar{X}-\bar{Y}-(\Delta_0)}{SE}<-t_{v,\alpha}$$`
--
Where 
- `$-z_\alpha$` and `$-t_{v,\alpha}$` are `$\small \alpha$` quantiles of standard normal and t-students with v degrees of freedom

---
### Two Samples - Mean - One sided - Case 1

.center[
<img src="data:image/png;base64,#C_3_slides_c_files/figure-html/unnamed-chunk-7-1.png" width="100%" />
]

- .blue[Intuition]: 
 - We never reject `$H_0$` if `$\bar{X}-\bar{Y} \geq \Delta_0$`.
 - We reject `$H_0$` if `$\bar{X}-\bar{Y}$` is sufficiently smaller than `$\Delta_0$`
 - If null is true ( `$\bar{X}-\bar{Y}=\Delta_0$` ) then we reject with probability
`$$\small P(\frac{\bar{X}-\bar{Y}-\Delta_0}{\sqrt{s^2_X/n+s^2_Y/ m}}<-z_\alpha)=P(Z<-z_\alpha)=\alpha$$`

---

### Two Samples - Mean - One sided - Case 2

**Hypothesis:**
`$H_0: \mu_X-\mu_Y=\Delta_0$` and 
`$H_A: \mu_X-\mu_Y>\Delta_0$`

- This includes any null hypothesis like `$\mu_X-\mu_Y \leq \Delta_0$`. Why?

--
- If you rejected `$\mu_X-\mu_Y=\Delta_0$` at `$\alpha$`, you would reject for sure anything smaller than `$\Delta_0$`

--
**Rejection Region:**
For a test of significance level `$\alpha$`, reject

- If large sample or known variance with normal
`$$\small \frac{\bar{X}-\bar{Y}-(\Delta_0)}{SE}>z_\alpha$$`
--

- If small sample from normal
`$$\small \frac{\bar{X}-\bar{Y}-(\Delta_0)}{SE}>t_{v,\alpha}$$`
--
Where 
- `$z_\alpha$` and `$t_{v,\alpha}$` are  `$\small (1-\alpha)$` quantiles of standard normal and t-students with v degrees of freedom

---
### Two Samples - Mean - One sided - Case 2

.center[
<img src="data:image/png;base64,#C_3_slides_c_files/figure-html/unnamed-chunk-8-1.png" width="100%" />
]

- .blue[Intuition]: 
  -  We never reject `$H_0$` if `$\bar{X}-\bar{Y} \leq \Delta_0$`.
  -  We reject `$H_0$` if `$\bar{X}-\bar{Y}$` is sufficiently larger than `$\Delta_0$`
  -  If null is true ( `$\bar{X}-\bar{Y}=\Delta_0$` ) then we reject with probability
`$$\small P(\frac{\bar{X}-\bar{Y}-\Delta_0}{\sqrt{s^2_X/n+s^2_Y/ m}}>z_\alpha)=P(Z>z_\alpha)=\alpha$$`

---
Is the price in dirty apartments the same as price in clean apartments?

|   Sample   |    n    | Sample Mean | Sample Standard Deviation |
|------------|---------|-------------|---------------------------|
|   Clean    |   100   |    1245     |            962            |
|   Dirty    |   100   |    869      |            693            |

- State the null and alternative hypothesis

--
- Calculate the value of the test statistic

--
- Calculate the p value

--
- At which `$\alpha$` we reject the null?

--
- Calculate 95\% confidence interval for the difference

---
We suspect Burger King might have larger caloric content for kids' meal. Let's test it.

|   Sample   |    n    | Sample Mean | Sample Standard Deviation |
|------------|---------|-------------|---------------------------|
|   McDonald    |   9    |    475     |           230            |
|   Burger King    |   16    |    520      |            180            |

- State the null and alternative hypothesis

--
- Calculate the value of the test statistic

--
- Determine the appropriate number of degrees of freedom

--
- Calculate the p value

--
- At which `$\alpha$` we reject the null?

---

### Comparing variances in two samples

**Example Problem 1:** is price dispersion the same in clean or dirty apartments?

**Example Problem 2:** is variance of prices of two stocks the same?

We need a test for variances in two different samples.

---
### Hypothesis testing with variances

- Suppose you have an iid sample `$X_1,X_2,...X_n$` and iid sample `$Y_1,Y_2,...Y_m$`, both from .blue[normal distributions]
- Denote `$s^2_1$` and `$s^2_2$` their sample variances and `$\sigma^2_1$` and `$\sigma^2_2$` their population variances

**Test Statistic** and its distribution under the null is:

Null hypothesis: `$\small H_A: \sigma_1 = \sigma_2$`
`$$F_{test}=\frac{s^2_1}{s^2_2} \sim F_{n-1,m-1}$$`

**Rejection region**
1. `$\small H_A: \sigma_1 \neq\sigma_2$`
 - Reject null if `$\small F_{test}<F_{\frac{\alpha}{2},n-1,m-1}$` or `$\small F_{test}>F_{1-\frac{\alpha}{2},n-1,m-1}$`

--
2. `$\small H_A: \sigma_1 \leq \sigma_2$`
 - Reject null if `$\small F_{test}<F_{\alpha,n-1,m-1}$`

--
3. `$\small H_A: \sigma_1 \geq \sigma_2$`
  - Reject null if `$\small F_{test}>F_{1-\alpha,n-1,m-1}$`

--
Where:
- `$\small F_{\alpha,n-1,m-1}$` is `$\small \alpha$` quantile of `$F_{n-1,m-1}$`
- So `$\small P(F<F_{\alpha,n-1,m-1})=\alpha$`
- `$\small F_{n-1,m-1}$` distribution is not symmetric around 0!
- So the notation is slightly different!!!!!

---
### F-distribution

If `$X_1$` and `$X_2$` are chi-square distributed and independent, then the ratio of `$X_1$` by its degrees of freedom to `$X_2$` by its degrees of freedom is distributed as F.

`$$F=\frac{X1/v_1}{X2/v_2} \sim F_{v1,v2}$$`
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2.030579e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.16290727 y: 2.339893e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.17543860 y: 2.645647e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.18796992 y: 2.944040e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.20050125 y: 3.232064e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.21303258 y: 3.507415e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.22556391 y: 3.768406e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.23809524 y: 4.013872e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.25062657 y: 4.243088e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.26315789 y: 4.455686e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.27568922 y: 4.651593e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.28822055 y: 4.830963e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.30075188 y: 4.994134e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.31328321 y: 5.141581e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.32581454 y: 5.273880e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.33834586 y: 5.391683e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.35087719 y: 5.495685e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.36340852 y: 5.586616e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.37593985 y: 5.665213e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.38847118 y: 5.732221e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.40100251 y: 5.788370e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.41353383 y: 5.834379e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.42606516 y: 5.870943e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.43859649 y: 5.898733e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.45112782 y: 5.918390e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.46365915 y: 5.930528e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.47619048 y: 5.935728e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.48872180 y: 5.934539e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.50125313 y: 5.927482e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.51378446 y: 5.915044e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.52631579 y: 5.897683e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.53884712 y: 5.875828e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.55137845 y: 5.849881e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.56390977 y: 5.820216e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.57644110 y: 5.787182e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.58897243 y: 5.751103e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.60150376 y: 5.712280e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.61403509 y: 5.670995e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.62656642 y: 5.627506e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.63909774 y: 5.582055e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.65162907 y: 5.534863e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.66416040 y: 5.486137e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.67669173 y: 5.436067e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.68922306 y: 5.384828e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.70175439 y: 5.332582e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.71428571 y: 5.279478e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.72681704 y: 5.225654e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.73934837 y: 5.171234e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 0.75187970 y: 5.116337e-01 distribution: F-distribution (df1 = 10 , df2 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3.785386e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.07769424 y: 3.737954e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.09022556 y: 3.691105e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.10275689 y: 3.644841e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.11528822 y: 3.599166e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.12781955 y: 3.554079e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.14035088 y: 3.509580e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.15288221 y: 3.465670e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.16541353 y: 3.422347e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.17794486 y: 3.379609e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.19047619 y: 3.337454e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.20300752 y: 3.295877e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.21553885 y: 3.254877e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.22807018 y: 3.214448e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.24060150 y: 3.174588e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.25313283 y: 3.135290e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.26566416 y: 3.096550e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.27819549 y: 3.058363e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.29072682 y: 3.020724e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.30325815 y: 2.983627e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.31578947 y: 2.947065e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.32832080 y: 2.911034e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.34085213 y: 2.875526e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.35338346 y: 2.840536e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.36591479 y: 2.806057e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.37844612 y: 2.772083e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.39097744 y: 2.738607e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.40350877 y: 2.705623e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.41604010 y: 2.673124e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.42857143 y: 2.641103e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.44110276 y: 2.609554e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.45363409 y: 2.578471e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.46616541 y: 2.547846e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.47869674 y: 2.517672e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.49122807 y: 2.487944e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.50375940 y: 2.458655e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.51629073 y: 2.429798e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.52882206 y: 2.401367e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.54135338 y: 2.373354e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.55388471 y: 2.345755e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.56641604 y: 2.318562e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.57894737 y: 2.291769e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.59147870 y: 2.265369e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.60401003 y: 2.239358e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.61654135 y: 2.213728e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.62907268 y: 2.188473e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.64160401 y: 2.163588e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.65413534 y: 2.139066e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.66666667 y: 2.114902e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.67919799 y: 2.091090e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.69172932 y: 2.067624e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.70426065 y: 2.044499e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.71679198 y: 2.021709e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.72932331 y: 1.999249e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.74185464 y: 1.977113e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.75438596 y: 1.955296e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.76691729 y: 1.933793e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.77944862 y: 1.912598e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.79197995 y: 1.891707e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.80451128 y: 1.871115e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.81704261 y: 1.850816e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.82957393 y: 1.830806e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.84210526 y: 1.811080e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.85463659 y: 1.791634e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.86716792 y: 1.772462e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.87969925 y: 1.753561e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.89223058 y: 1.734926e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.90476190 y: 1.716552e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.91729323 y: 1.698435e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.92982456 y: 1.680571e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.94235589 y: 1.662956e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.95488722 y: 1.645586e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.96741855 y: 1.628457e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.97994987 y: 1.611564e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 1.99248120 y: 1.594904e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.00501253 y: 1.578473e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.01754386 y: 1.562267e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.03007519 y: 1.546283e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.04260652 y: 1.530517e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.05513784 y: 1.514966e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.06766917 y: 1.499625e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.08020050 y: 1.484492e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.09273183 y: 1.469563e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.10526316 y: 1.454835e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.11779449 y: 1.440305e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.13032581 y: 1.425969e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.14285714 y: 1.411824e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.15538847 y: 1.397867e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.16791980 y: 1.384096e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.18045113 y: 1.370507e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.19298246 y: 1.357097e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.20551378 y: 1.343863e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.21804511 y: 1.330803e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.23057644 y: 1.317914e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.24310777 y: 1.305193e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.25563910 y: 1.292637e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.26817043 y: 1.280245e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.28070175 y: 1.268012e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.29323308 y: 1.255937e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.30576441 y: 1.244017e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.31829574 y: 1.232250e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.33082707 y: 1.220634e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.34335840 y: 1.209165e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.35588972 y: 1.197843e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.36842105 y: 1.186663e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.38095238 y: 1.175625e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.39348371 y: 1.164726e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.40601504 y: 1.153964e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.41854637 y: 1.143336e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.43107769 y: 1.132842e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.44360902 y: 1.122477e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.45614035 y: 1.112242e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.46867168 y: 1.102133e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.48120301 y: 1.092149e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.49373434 y: 1.082288e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.50626566 y: 1.072547e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.51879699 y: 1.062926e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.53132832 y: 1.053422e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.54385965 y: 1.044034e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.55639098 y: 1.034760e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.56892231 y: 1.025597e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.58145363 y: 1.016546e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.59398496 y: 1.007602e-01 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.60651629 y: 9.987665e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.61904762 y: 9.900362e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.63157895 y: 9.814098e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.64411028 y: 9.728860e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.65664160 y: 9.644630e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.66917293 y: 9.561396e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.68170426 y: 9.479141e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.69423559 y: 9.397853e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.70676692 y: 9.317516e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.71929825 y: 9.238116e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.73182957 y: 9.159641e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.74436090 y: 9.082077e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.75689223 y: 9.005410e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.76942356 y: 8.929627e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.78195489 y: 8.854717e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.79448622 y: 8.780666e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.80701754 y: 8.707462e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.81954887 y: 8.635093e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.83208020 y: 8.563547e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.84461153 y: 8.492812e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.85714286 y: 8.422877e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.86967419 y: 8.353731e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.88220551 y: 8.285362e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.89473684 y: 8.217759e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.90726817 y: 8.150913e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.91979950 y: 8.084811e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.93233083 y: 8.019445e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.94486216 y: 7.954802e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.95739348 y: 7.890875e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.96992481 y: 7.827652e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.98245614 y: 7.765123e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 2.99498747 y: 7.703280e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.00751880 y: 7.642113e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.02005013 y: 7.581613e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.03258145 y: 7.521770e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.04511278 y: 7.462575e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.05764411 y: 7.404020e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.07017544 y: 7.346096e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.08270677 y: 7.288795e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.09523810 y: 7.232108e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.10776942 y: 7.176026e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.12030075 y: 7.120542e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.13283208 y: 7.065648e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.14536341 y: 7.011335e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.15789474 y: 6.957597e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.17042607 y: 6.904425e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.18295739 y: 6.851811e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.19548872 y: 6.799750e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.20802005 y: 6.748232e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.22055138 y: 6.697252e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.23308271 y: 6.646801e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.24561404 y: 6.596874e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.25814536 y: 6.547462e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.27067669 y: 6.498561e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.28320802 y: 6.450162e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.29573935 y: 6.402259e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.30827068 y: 6.354847e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.32080201 y: 6.307918e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.33333333 y: 6.261466e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.34586466 y: 6.215486e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.35839599 y: 6.169971e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.37092732 y: 6.124916e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.38345865 y: 6.080313e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.39598997 y: 6.036159e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.40852130 y: 5.992446e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.42105263 y: 5.949170e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.43358396 y: 5.906325e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.44611529 y: 5.863906e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.45864662 y: 5.821906e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.47117794 y: 5.780322e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.48370927 y: 5.739147e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.49624060 y: 5.698377e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.50877193 y: 5.658006e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.52130326 y: 5.618030e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.53383459 y: 5.578444e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.54636591 y: 5.539243e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.55889724 y: 5.500423e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.57142857 y: 5.461977e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.58395990 y: 5.423903e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.59649123 y: 5.386195e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.60902256 y: 5.348850e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.62155388 y: 5.311862e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.63408521 y: 5.275227e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.64661654 y: 5.238941e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.65914787 y: 5.203000e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.67167920 y: 5.167399e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.68421053 y: 5.132135e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.69674185 y: 5.097204e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.70927318 y: 5.062601e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.72180451 y: 5.028323e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.73433584 y: 4.994366e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.74686717 y: 4.960725e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.75939850 y: 4.927398e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.77192982 y: 4.894380e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.78446115 y: 4.861669e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.79699248 y: 4.829259e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.80952381 y: 4.797149e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.82205514 y: 4.765333e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.83458647 y: 4.733810e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.84711779 y: 4.702574e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.85964912 y: 4.671624e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.87218045 y: 4.640956e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.88471178 y: 4.610566e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.89724311 y: 4.580452e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.90977444 y: 4.550610e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.92230576 y: 4.521037e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.93483709 y: 4.491729e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.94736842 y: 4.462685e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.95989975 y: 4.433901e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.97243108 y: 4.405373e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.98496241 y: 4.377100e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 3.99749373 y: 4.349078e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.01002506 y: 4.321304e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.02255639 y: 4.293776e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.03508772 y: 4.266491e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.04761905 y: 4.239445e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.06015038 y: 4.212638e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.07268170 y: 4.186064e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.08521303 y: 4.159724e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.09774436 y: 4.133612e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.11027569 y: 4.107728e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.12280702 y: 4.082068e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.13533835 y: 4.056630e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.14786967 y: 4.031412e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.16040100 y: 4.006411e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.17293233 y: 3.981625e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.18546366 y: 3.957052e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.19799499 y: 3.932688e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.21052632 y: 3.908533e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.22305764 y: 3.884583e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.23558897 y: 3.860836e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.24812030 y: 3.837291e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.26065163 y: 3.813945e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.27318296 y: 3.790796e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.28571429 y: 3.767841e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.29824561 y: 3.745079e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.31077694 y: 3.722508e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.32330827 y: 3.700125e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.33583960 y: 3.677929e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.34837093 y: 3.655918e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.36090226 y: 3.634089e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.37343358 y: 3.612441e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.38596491 y: 3.590972e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.39849624 y: 3.569680e-02 distribution: F-distribution (df1 = 10 , df2 = 3)","x: 4.41102757 y: 3.548563e-02 distribution: F-distribution (df1 = 10 , df2 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0.90225564 y: 5.011004e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 0.91478697 y: 4.952341e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 0.92731830 y: 4.893704e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 0.93984962 y: 4.835163e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 0.95238095 y: 4.776782e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 0.96491228 y: 4.718617e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 0.97744361 y: 4.660720e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 0.98997494 y: 4.603139e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.00250627 y: 4.545917e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.01503759 y: 4.489090e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.02756892 y: 4.432693e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.04010025 y: 4.376757e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.05263158 y: 4.321309e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.06516291 y: 4.266374e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.07769424 y: 4.211971e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.09022556 y: 4.158121e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.10275689 y: 4.104839e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.11528822 y: 4.052139e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.12781955 y: 4.000034e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.14035088 y: 3.948533e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.15288221 y: 3.897645e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.16541353 y: 3.847377e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.17794486 y: 3.797734e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.19047619 y: 3.748720e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.20300752 y: 3.700339e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.21553885 y: 3.652592e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.22807018 y: 3.605479e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.24060150 y: 3.559001e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.25313283 y: 3.513158e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.26566416 y: 3.467946e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.27819549 y: 3.423363e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.29072682 y: 3.379408e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.30325815 y: 3.336076e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.31578947 y: 3.293363e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.32832080 y: 3.251264e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.34085213 y: 3.209775e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.35338346 y: 3.168890e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.36591479 y: 3.128604e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.37844612 y: 3.088911e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.39097744 y: 3.049804e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.40350877 y: 3.011277e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.41604010 y: 2.973325e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.42857143 y: 2.935939e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.44110276 y: 2.899113e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.45363409 y: 2.862840e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.46616541 y: 2.827113e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.47869674 y: 2.791925e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.49122807 y: 2.757268e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.50375940 y: 2.723136e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.51629073 y: 2.689520e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.52882206 y: 2.656414e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.54135338 y: 2.623810e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.55388471 y: 2.591700e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.56641604 y: 2.560079e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.57894737 y: 2.528937e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.59147870 y: 2.498268e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.60401003 y: 2.468065e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.61654135 y: 2.438320e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.62907268 y: 2.409026e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.64160401 y: 2.380176e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.65413534 y: 2.351763e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.66666667 y: 2.323781e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.67919799 y: 2.296221e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.69172932 y: 2.269078e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.70426065 y: 2.242345e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.71679198 y: 2.216014e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.72932331 y: 2.190079e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.74185464 y: 2.164535e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.75438596 y: 2.139373e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.76691729 y: 2.114589e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.77944862 y: 2.090175e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.79197995 y: 2.066126e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.80451128 y: 2.042435e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.81704261 y: 2.019096e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.82957393 y: 1.996104e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.84210526 y: 1.973453e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.85463659 y: 1.951136e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.86716792 y: 1.929149e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.87969925 y: 1.907486e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.89223058 y: 1.886141e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.90476190 y: 1.865108e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.91729323 y: 1.844384e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.92982456 y: 1.823961e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.94235589 y: 1.803836e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.95488722 y: 1.784004e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.96741855 y: 1.764458e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.97994987 y: 1.745195e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 1.99248120 y: 1.726210e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.00501253 y: 1.707497e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.01754386 y: 1.689053e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.03007519 y: 1.670873e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.04260652 y: 1.652952e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.05513784 y: 1.635286e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.06766917 y: 1.617871e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.08020050 y: 1.600702e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.09273183 y: 1.583776e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.10526316 y: 1.567088e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.11779449 y: 1.550634e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.13032581 y: 1.534410e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.14285714 y: 1.518413e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.15538847 y: 1.502638e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.16791980 y: 1.487083e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.18045113 y: 1.471743e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.19298246 y: 1.456614e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.20551378 y: 1.441694e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.21804511 y: 1.426978e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.23057644 y: 1.412464e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.24310777 y: 1.398148e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.25563910 y: 1.384026e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.26817043 y: 1.370097e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.28070175 y: 1.356355e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.29323308 y: 1.342799e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.30576441 y: 1.329425e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.31829574 y: 1.316230e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.33082707 y: 1.303212e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.34335840 y: 1.290367e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.35588972 y: 1.277693e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.36842105 y: 1.265187e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.38095238 y: 1.252846e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.39348371 y: 1.240668e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.40601504 y: 1.228649e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.41854637 y: 1.216788e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.43107769 y: 1.205082e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.44360902 y: 1.193528e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.45614035 y: 1.182123e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.46867168 y: 1.170867e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.48120301 y: 1.159755e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.49373434 y: 1.148787e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.50626566 y: 1.137959e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.51879699 y: 1.127269e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.53132832 y: 1.116716e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.54385965 y: 1.106296e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.55639098 y: 1.096009e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.56892231 y: 1.085851e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.58145363 y: 1.075822e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.59398496 y: 1.065918e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.60651629 y: 1.056138e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.61904762 y: 1.046480e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.63157895 y: 1.036943e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.64411028 y: 1.027523e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.65664160 y: 1.018220e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.66917293 y: 1.009032e-01 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.68170426 y: 9.999566e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.69423559 y: 9.909925e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.70676692 y: 9.821379e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.71929825 y: 9.733911e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.73182957 y: 9.647505e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.74436090 y: 9.562145e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.75689223 y: 9.477816e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.76942356 y: 9.394502e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.78195489 y: 9.312187e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.79448622 y: 9.230858e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.80701754 y: 9.150498e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.81954887 y: 9.071094e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.83208020 y: 8.992631e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.84461153 y: 8.915096e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.85714286 y: 8.838475e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.86967419 y: 8.762755e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.88220551 y: 8.687921e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.89473684 y: 8.613962e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.90726817 y: 8.540865e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.91979950 y: 8.468616e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.93233083 y: 8.397204e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.94486216 y: 8.326617e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.95739348 y: 8.256842e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.96992481 y: 8.187868e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.98245614 y: 8.119683e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 2.99498747 y: 8.052277e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.00751880 y: 7.985638e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.02005013 y: 7.919754e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.03258145 y: 7.854616e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.04511278 y: 7.790213e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.05764411 y: 7.726534e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.07017544 y: 7.663569e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.08270677 y: 7.601309e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.09523810 y: 7.539742e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.10776942 y: 7.478860e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.12030075 y: 7.418653e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.13283208 y: 7.359112e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 3.14536341 y: 7.300226e-02 distribution: F-distribution (df1 = 100 , df2 = 3)","x: 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0.91478697 y: 4.397823e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 0.92731830 y: 4.343874e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 0.93984962 y: 4.290437e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 0.95238095 y: 4.237517e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 0.96491228 y: 4.185120e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 0.97744361 y: 4.133251e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 0.98997494 y: 4.081913e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.00250627 y: 4.031110e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.01503759 y: 3.980845e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.02756892 y: 3.931119e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.04010025 y: 3.881934e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.05263158 y: 3.833290e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.06516291 y: 3.785189e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.07769424 y: 3.737630e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.09022556 y: 3.690612e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.10275689 y: 3.644135e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.11528822 y: 3.598198e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.12781955 y: 3.552799e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.14035088 y: 3.507936e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.15288221 y: 3.463607e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.16541353 y: 3.419809e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.17794486 y: 3.376541e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.19047619 y: 3.333798e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.20300752 y: 3.291579e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.21553885 y: 3.249879e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.22807018 y: 3.208695e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.24060150 y: 3.168023e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.25313283 y: 3.127860e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.26566416 y: 3.088201e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.27819549 y: 3.049043e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.29072682 y: 3.010382e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.30325815 y: 2.972212e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.31578947 y: 2.934530e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.32832080 y: 2.897331e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.34085213 y: 2.860611e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.35338346 y: 2.824365e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.36591479 y: 2.788588e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.37844612 y: 2.753276e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.39097744 y: 2.718423e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.40350877 y: 2.684026e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.41604010 y: 2.650078e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.42857143 y: 2.616576e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.44110276 y: 2.583515e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.45363409 y: 2.550889e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.46616541 y: 2.518693e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.47869674 y: 2.486923e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.49122807 y: 2.455574e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.50375940 y: 2.424641e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.51629073 y: 2.394119e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.52882206 y: 2.364003e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.54135338 y: 2.334288e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.55388471 y: 2.304970e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.56641604 y: 2.276043e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.57894737 y: 2.247503e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.59147870 y: 2.219344e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.60401003 y: 2.191563e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.61654135 y: 2.164154e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.62907268 y: 2.137112e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.64160401 y: 2.110434e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.65413534 y: 2.084113e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.66666667 y: 2.058147e-01 distribution: F-distribution (df1 = 3 , df2 = 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1.82957393 y: 1.750729e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.84210526 y: 1.729250e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.85463659 y: 1.708059e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.86716792 y: 1.687154e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.87969925 y: 1.666531e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.89223058 y: 1.646184e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.90476190 y: 1.626111e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.91729323 y: 1.606308e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.92982456 y: 1.586771e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.94235589 y: 1.567496e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.95488722 y: 1.548480e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.96741855 y: 1.529719e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.97994987 y: 1.511210e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 1.99248120 y: 1.492949e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.00501253 y: 1.474932e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.01754386 y: 1.457156e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.03007519 y: 1.439618e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.04260652 y: 1.422315e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.05513784 y: 1.405243e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.06766917 y: 1.388398e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.08020050 y: 1.371779e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.09273183 y: 1.355381e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.10526316 y: 1.339201e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.11779449 y: 1.323237e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.13032581 y: 1.307485e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.14285714 y: 1.291942e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.15538847 y: 1.276606e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.16791980 y: 1.261473e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.18045113 y: 1.246541e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.19298246 y: 1.231807e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.20551378 y: 1.217267e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.21804511 y: 1.202920e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.23057644 y: 1.188762e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.24310777 y: 1.174791e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.25563910 y: 1.161005e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.26817043 y: 1.147400e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.28070175 y: 1.133974e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.29323308 y: 1.120724e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.30576441 y: 1.107648e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.31829574 y: 1.094744e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.33082707 y: 1.082009e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.34335840 y: 1.069441e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.35588972 y: 1.057037e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.36842105 y: 1.044795e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.38095238 y: 1.032713e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.39348371 y: 1.020788e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.40601504 y: 1.009019e-01 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.41854637 y: 9.974027e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.43107769 y: 9.859373e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.44360902 y: 9.746207e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.45614035 y: 9.634508e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.46867168 y: 9.524255e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.48120301 y: 9.415428e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.49373434 y: 9.308007e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.50626566 y: 9.201972e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.51879699 y: 9.097303e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.53132832 y: 8.993981e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.54385965 y: 8.891988e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.55639098 y: 8.791305e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.56892231 y: 8.691913e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.58145363 y: 8.593795e-02 distribution: F-distribution (df1 = 3 , df2 = 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2.74436090 y: 7.426345e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.75689223 y: 7.344292e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.76942356 y: 7.263271e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.78195489 y: 7.183267e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.79448622 y: 7.104266e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.80701754 y: 7.026254e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.81954887 y: 6.949218e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.83208020 y: 6.873143e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.84461153 y: 6.798018e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.85714286 y: 6.723828e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.86967419 y: 6.650561e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.88220551 y: 6.578204e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.89473684 y: 6.506745e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.90726817 y: 6.436172e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.91979950 y: 6.366471e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.93233083 y: 6.297632e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.94486216 y: 6.229642e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.95739348 y: 6.162489e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.96992481 y: 6.096164e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.98245614 y: 6.030653e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 2.99498747 y: 5.965946e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.00751880 y: 5.902033e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.02005013 y: 5.838901e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.03258145 y: 5.776542e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.04511278 y: 5.714943e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.05764411 y: 5.654095e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.07017544 y: 5.593988e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.08270677 y: 5.534612e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.09523810 y: 5.475956e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.10776942 y: 5.418011e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.12030075 y: 5.360768e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.13283208 y: 5.304217e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.14536341 y: 5.248349e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.15789474 y: 5.193154e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.17042607 y: 5.138623e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.18295739 y: 5.084748e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.19548872 y: 5.031520e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.20802005 y: 4.978930e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.22055138 y: 4.926970e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.23308271 y: 4.875631e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.24561404 y: 4.824905e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.25814536 y: 4.774784e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.27067669 y: 4.725259e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.28320802 y: 4.676323e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.29573935 y: 4.627968e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.30827068 y: 4.580186e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.32080201 y: 4.532971e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.33333333 y: 4.486313e-02 distribution: F-distribution (df1 = 3 , df2 = 10)","x: 3.34586466 y: 4.440206e-02 distribution: F-distribution 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distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.62656642 y: 4.683907e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.63909774 y: 4.622844e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.65162907 y: 4.562499e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.66416040 y: 4.502887e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.67669173 y: 4.444019e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.68922306 y: 4.385903e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.70175439 y: 4.328546e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.71428571 y: 4.271953e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.72681704 y: 4.216127e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.73934837 y: 4.161069e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.75187970 y: 4.106780e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.76441103 y: 4.053258e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.77694236 y: 4.000502e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.78947368 y: 3.948508e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.80200501 y: 3.897272e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.81453634 y: 3.846789e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.82706767 y: 3.797054e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.83959900 y: 3.748060e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.85213033 y: 3.699802e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.86466165 y: 3.652272e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.87719298 y: 3.605462e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.88972431 y: 3.559365e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.90225564 y: 3.513972e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.91478697 y: 3.469275e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.92731830 y: 3.425265e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.93984962 y: 3.381934e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.95238095 y: 3.339272e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.96491228 y: 3.297270e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.97744361 y: 3.255919e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 0.98997494 y: 3.215210e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.00250627 y: 3.175134e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.01503759 y: 3.135680e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.02756892 y: 3.096840e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.04010025 y: 3.058604e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.05263158 y: 3.020962e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.06516291 y: 2.983906e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.07769424 y: 2.947427e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.09022556 y: 2.911514e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.10275689 y: 2.876158e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.11528822 y: 2.841352e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.12781955 y: 2.807085e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.14035088 y: 2.773348e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.15288221 y: 2.740134e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.16541353 y: 2.707432e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.17794486 y: 2.675235e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.19047619 y: 2.643534e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.20300752 y: 2.612320e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.21553885 y: 2.581585e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.22807018 y: 2.551321e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.24060150 y: 2.521520e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.25313283 y: 2.492174e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.26566416 y: 2.463274e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.27819549 y: 2.434814e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.29072682 y: 2.406785e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.30325815 y: 2.379181e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.31578947 y: 2.351993e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.32832080 y: 2.325215e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.34085213 y: 2.298839e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.35338346 y: 2.272859e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.36591479 y: 2.247267e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.37844612 y: 2.222056e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.39097744 y: 2.197221e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.40350877 y: 2.172754e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.41604010 y: 2.148649e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.42857143 y: 2.124900e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.44110276 y: 2.101500e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.45363409 y: 2.078444e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.46616541 y: 2.055725e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.47869674 y: 2.033338e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.49122807 y: 2.011276e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.50375940 y: 1.989535e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.51629073 y: 1.968108e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.52882206 y: 1.946990e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.54135338 y: 1.926176e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.55388471 y: 1.905661e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.56641604 y: 1.885439e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.57894737 y: 1.865505e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.59147870 y: 1.845855e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.60401003 y: 1.826483e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.61654135 y: 1.807385e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.62907268 y: 1.788556e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.64160401 y: 1.769991e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.65413534 y: 1.751687e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.66666667 y: 1.733637e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.67919799 y: 1.715839e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.69172932 y: 1.698288e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.70426065 y: 1.680980e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.71679198 y: 1.663910e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.72932331 y: 1.647075e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.74185464 y: 1.630470e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.75438596 y: 1.614092e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.76691729 y: 1.597937e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.77944862 y: 1.582001e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.79197995 y: 1.566281e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.80451128 y: 1.550773e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.81704261 y: 1.535474e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.82957393 y: 1.520379e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.84210526 y: 1.505486e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.85463659 y: 1.490792e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.86716792 y: 1.476293e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.87969925 y: 1.461985e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.89223058 y: 1.447866e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.90476190 y: 1.433933e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.91729323 y: 1.420182e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.92982456 y: 1.406612e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.94235589 y: 1.393217e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.95488722 y: 1.379997e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.96741855 y: 1.366948e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.97994987 y: 1.354068e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 1.99248120 y: 1.341353e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.00501253 y: 1.328801e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.01754386 y: 1.316409e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.03007519 y: 1.304175e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.04260652 y: 1.292097e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.05513784 y: 1.280172e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.06766917 y: 1.268397e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.08020050 y: 1.256770e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.09273183 y: 1.245289e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.10526316 y: 1.233951e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.11779449 y: 1.222755e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.13032581 y: 1.211698e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.14285714 y: 1.200778e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.15538847 y: 1.189993e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.16791980 y: 1.179340e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.18045113 y: 1.168819e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.19298246 y: 1.158425e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.20551378 y: 1.148159e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.21804511 y: 1.138017e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.23057644 y: 1.127999e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.24310777 y: 1.118101e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.25563910 y: 1.108323e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.26817043 y: 1.098661e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.28070175 y: 1.089116e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.29323308 y: 1.079684e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.30576441 y: 1.070365e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.31829574 y: 1.061156e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.33082707 y: 1.052056e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.34335840 y: 1.043063e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.35588972 y: 1.034176e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.36842105 y: 1.025393e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.38095238 y: 1.016713e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.39348371 y: 1.008133e-01 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.40601504 y: 9.996538e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.41854637 y: 9.912721e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.43107769 y: 9.829872e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.44360902 y: 9.747976e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.45614035 y: 9.667019e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.46867168 y: 9.586988e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.48120301 y: 9.507870e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.49373434 y: 9.429650e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.50626566 y: 9.352317e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.51879699 y: 9.275857e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.53132832 y: 9.200259e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.54385965 y: 9.125510e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.55639098 y: 9.051598e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.56892231 y: 8.978511e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.58145363 y: 8.906238e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.59398496 y: 8.834766e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.60651629 y: 8.764086e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.61904762 y: 8.694186e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.63157895 y: 8.625054e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.64411028 y: 8.556681e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.65664160 y: 8.489056e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.66917293 y: 8.422169e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.68170426 y: 8.356008e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.69423559 y: 8.290565e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.70676692 y: 8.225830e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.71929825 y: 8.161792e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.73182957 y: 8.098443e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.74436090 y: 8.035772e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.75689223 y: 7.973772e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.76942356 y: 7.912432e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.78195489 y: 7.851744e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.79448622 y: 7.791699e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.80701754 y: 7.732288e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.81954887 y: 7.673504e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.83208020 y: 7.615337e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.84461153 y: 7.557779e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.85714286 y: 7.500823e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.86967419 y: 7.444460e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.88220551 y: 7.388683e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.89473684 y: 7.333483e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.90726817 y: 7.278854e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.91979950 y: 7.224787e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.93233083 y: 7.171276e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.94486216 y: 7.118313e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.95739348 y: 7.065892e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.96992481 y: 7.014004e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.98245614 y: 6.962643e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 2.99498747 y: 6.911803e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.00751880 y: 6.861477e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.02005013 y: 6.811657e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.03258145 y: 6.762339e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.04511278 y: 6.713514e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.05764411 y: 6.665178e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.07017544 y: 6.617323e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.08270677 y: 6.569944e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.09523810 y: 6.523035e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.10776942 y: 6.476589e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.12030075 y: 6.430601e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.13283208 y: 6.385066e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.14536341 y: 6.339977e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.15789474 y: 6.295330e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.17042607 y: 6.251117e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.18295739 y: 6.207335e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.19548872 y: 6.163978e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.20802005 y: 6.121040e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.22055138 y: 6.078517e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 3.23308271 y: 6.036403e-02 distribution: F-distribution (df1 = 3 , df2 = 3)","x: 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---
### Variance test and F-distribution

How come under the null we get F statistics? `$s^2$` is not chi square.... but!

`$$\frac{(n-1)s^2_1}{\sigma^2_1} \sim \chi_{n-1}$$`

`$$F_{test}=\frac{\frac{(n-1)s^2_1}{\sigma^2_1(n-1)}}{\frac{(m-1)s^2_2}{\sigma^2_2(m-1)}} \sim F_{n-1,m-1}$$`

Under null we have that `$\sigma_1=\sigma_2$`, so

`$$F_{test}=\frac{\frac{(n-1)s^2_1}{\sigma^2_1(n-1)}}{\frac{(m-1)s^2_2}{\sigma^2_2(m-1)}}=\frac{s^2_1}{s^2_2}$$`

---
**Example:**

Let's test the equality of variances of clean and dirty apartments. 
- `$H_0: \sigma^2_c=\sigma^2_d$`
- `$H_A: \sigma^2_c \neq \sigma^2_d$`

- Variances in each samples are: `$s^2_c=925255.5$` and `$s^2_d=480681.9$`

- We have 100 observations in each sample, so 99 degrees of freedom on each side

`$F_{test}=\frac{s^2_c}{s^2_d}=\frac{925255.5}{480681.9}=1.924$`

Critical regions at 5% are:
- `$\small F_{\frac{\alpha}{2},n-1,m-1}=F_{0.025,99,99}=0.67284$`
- `$\small F_{1-\frac{\alpha}{2},n-1,m-1}=F_{0.975,99,99}=1.48623$`

--
- We reject the equality at 5%

---
### Exercise

The time C is a commute time of itam students distributed as C ∼ N(30, 10). Two random samples of commute times are taken for two days of sizes `$n_1 = 15$` and `$n_2 = 18$` students. They both come from the same distribution specified above. If sample variances `$s_1^2$` and `$s_2^2$` are to be reported,  What is the probability that variance in sample one is more than twice as big as sample on day 2?

--
`$$\small \frac{(n-1)s^2_1}{\sigma^2_1} \sim \chi_{n-1}$$`
`$$\small F_{test}=\frac{\frac{(n-1)s^2_1}{\sigma^2_1(n-1)}}{\frac{(m-1)s^2_2}{\sigma^2_2(m-1)}} \sim F_{n-1,m-1}$$`

- They are from the same distribution so `$\sigma_1=\sigma_2$`

`$P(s_1^2>2s_2^2)=P(\frac{s_1^2}{s_2^2}>2)=P(F_{14,17}>2)$`

---
layout: false
class: inverse, middle

# Hypothesis Testing: Paired Data

---

### Paired Data

- Sometimes we have only one set of individuals or objects, but two (or more) observations per each.

- .blue[Example]: We look at employees productivity. For each employee we have two observations: their productivity when working from home and their productivity when working from the office.

- .blue[Example]: We look at sales at Walmart and Soriana in each neighborhood. We have two observations per neighborhood: sales at local Walmart and sales at local Soriana

---
### Paired Data

We have a set of independent pairs of data `$(X_1,Y_1),(X_2,Y_2),..., (X_n,Y_n)$`

--
Observation within the pair are not necessarily independent!
- Some productive workers are productive both in the office and at home, some unproductive are unproductive both in the office and at home.

- We are interested in the difference between `$X$` and `$Y$`. We observe that difference for each individual in the sample: `$X_i-Y_i=d_i$`

- We want to look at the population value of that difference: `$\mu_X-\mu_Y=\Delta$`
  - We can check if the means are the same: `$\Delta=0$`
      - No difference in productivity between work at home and at the office
  - Or mean `$X$` is bigger than mean `$Y$`:  `$\Delta>0$`

---
### Paired Data

Null hypothesis: `$H_0: \Delta=\Delta_0$`

**Test Statistic** is:

`$$\text{test statistic}=\frac{\overbrace{\bar{X}-\bar{Y}}^{\bar{d}}-\Delta_0}{SE}=\frac{\bar{d}-\Delta_0}{SE}$$`

--
- Note on the Standard Error - it's not the same formula as for the two independent samples:
`$Var(X_i-Y_i) \neq Var(X_i)+Var(Y_i)$`

- We just calculate it from the differences. 
  - If we know the variance, `$SE=\frac{\sqrt{var(d_i)}}{\sqrt n}=\frac{\sigma_d}{\sqrt n}$`
  - If we don't: `$SE=\frac{s_d}{\sqrt n}$`

---
### Paired Data

**Rejection Region:**
For a test of significance level `$\alpha$`:

- If large sample or known variance with normal distribution:

- for `$H_A: \bar{d} \neq \Delta_0$`, reject if:
`$$\small \frac{\bar{d}-\Delta_0}{SE}<-z_\frac{\alpha}{2} \qquad or \qquad \frac{\bar{d}-\Delta_0}{SE}>z_\frac{\alpha}{2}$$`
--

- for `$H_A: \bar{d} > \Delta_0$`, reject if:
`$$\small \frac{\bar{d}-\Delta_0}{SE}>z_\alpha$$`
--

- for `$H_A: \bar{d} < \Delta_0$`, reject if:
`$$\small \frac{\bar{d}-\Delta_0}{SE}<-z_\alpha$$`
--

- If small sample from normal, replace critical values with .blue[t-distribution with n-1 degrees of freedom]

---
### Paired or Unpaired Data?

1. We would like to know if Intel’s stock and Southwest Airlines’
stock have similar rates of return. To find out, we take a
random sample of 50 days, and record Intel’s and Southwest’s
stock on those same days

--
- Paired

--
2. We randomly sample 50 items from Target stores and note
the price for each. Then we visit Walmart and collect the price
for each of those same 50 items.

--
- Paired

--
3. A school board would like to determine whether there is a
difference in average SAT scores for students at one high
school versus another high school in the district. To check,
they take a simple random sample of 100 students from each
high school

--
- Not paired

---

A psychologist thinks that age influences IQ. They take a random sample of 100 people of age 40. For each person we know their IQ at age 16 and now. On average, in this sample, IQ at young age was 8 points higher than at age 40. Standard deviation of that difference was 7 points. Using `$\alpha=0.01$` test the hypothesis that IQ decreases with age.

---
#### Hypothesis Testing for Correlation

.blue[Example]: are cleanliness score (C) and price (P) correlated in the population?

Assume you have **normally distributed variables** in **independent pairs**. `$\{(C_1,P_1), (C_2, P_2),...\}$`

.blue[Null hypothesis]: `$H_0: \rho(C,P)=\rho_0$`

**Test statistic** and its distribution under the null:

`$$\small T_{test}=\frac{\hat{\rho}(C,P)-\rho_0}{\sqrt{\frac{1-\hat{\rho}(C,P)^2}{n-2}}} \sim t_{n-2}$$`

where `$\hat{\rho}(C,P)$` is sample correlation coefficient

.blue[Alternative hypothesis] and their **rejection regions**:
- `$\small H_A: \rho(C,P) \neq \rho_0$` - reject `$\small H_0$` if `$\small t_{test}>t_{\frac{\alpha}{2},n-2}$` or `$\small t_{test}<-t_{\frac{\alpha}{2},n-2}$`
- `$\small H_A: \rho(C,P)>\rho_0$` - reject `$\small H_0$` if `$\small t_{test}>t_{\alpha,n-2}$`
- `$\small H_A: \rho(C,P)<\rho_0$` - reject `$\small H_0$` if `$\small t_{test}<-t_{\alpha,n-2}$`

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### Correlation test

- Suppose we test `$H_0: \rho(C,P)=0$` vs `$\rho(C,P)>0$` in our Airbnb Data (clean score vs price)

- We **assume** that:
  - The variables are normally distributed
  - Pairs are independent
  - Relationship would be linear

- Sample correlation is: 0.159, we have `$n=200$`

- So `$t_{test}=\frac{\hat{\rho}(C,P)\sqrt {(n-2)}}{\sqrt{1-\hat{\rho}(C,P)^2}}=\frac{0.159\sqrt {198}}{\sqrt{1-0.025}}=2.27$`

- Since `$n=200$`, t-student is identical to standard normal. We can either
  - Compare calculated statistic to critical regions at various significance levels
  - Compute its p-value
  
- `$p-value=P(Z>t_{test})=P(Z>2.27)=0.0126$`

--
  - Test at `$\alpha=0.05$` would reject `$H_0$`

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### Correlation Test

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### Correlation Test

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### Exercise
Midterm 2, fall 2023, Q8

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### Exercise

Suppose you have a random sample of 27 units (from a bivariate normal). X measures client's age and Y measures their spending.  You calculated the correlation coefficient of `$\hat{\rho}=-0.45$`. Can you reject null of `$\rho=0$` in favor of alternative `$\rho \neq 0$` at 5% significance level?

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### Exercises
- Review Exercises:
  - All the remaining from PDFs 4 and 5

- Homeworks
  - All the remaining from Lista 00.2